# Table of thermodynamic equations

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$c=$
$T$$\partial S$$N$$\partial T$
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$\beta =-$
$1$$\partial V$$V$$\partial p$
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$\alpha =$
$1$$\partial V$$V$$\partial T$ :
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vte :
Common thermodynamic equations and quantities in thermodynamics, using mathematical notation, are as follows:

## Definitions

Many of the definitions below are also used in the thermodynamics of chemical reactions.

### General basic quantities

| Quantity (common name/s) | (Common) symbol/s | SI unit | Dimension |
| Number of molecules | N | 1 | 1 |
| Amount of substance | n | mol | N |
| Temperature | T | K | Θ |
| Heat Energy | Q, q | J | ML$^2$T$^{−2}$ |
| Latent heat | Q$_L$ | J | ML$^2$T$^{−2}$ |

### General derived quantities

| Quantity (common name/s) | (Common) symbol/s | Defining equation | SI unit | Dimension |
| Thermodynamic beta, inverse temperature | β | $\beta =1/k_{\text{B}}T$ | J$^{−1}$ | T$^2$M$^{−1}$L$^{−2}$ |
| Thermodynamic temperature | τ | $\tau =k_{\text{B}}T$ $\tau =k_{\text{B}}\left(\partial U/\partial S\right)_{N}$ $1/\tau =1/k_{\text{B}}\left(\partial S/\partial U\right)_{N}$ | J | ML$^2$T$^{−2}$ |
| Entropy | S | $S=-k_{\text{B}}\sum _{i}p_{i}\ln p_{i}$ $S=-\left(\partial F/\partial T\right)_{V,N}$ , $S=-\left(\partial G/\partial T\right)_{P,N}$ | J⋅K$^{−1}$ | ML$^2$T$^{−2}$Θ$^{−1}$ |
| Pressure | P | $P=-\left(\partial F/\partial V\right)_{T,N}$ $P=-\left(\partial U/\partial V\right)_{S,N}$ | Pa | ML$^{−1}$T$^{−2}$ |
| Internal Energy | U | $U=\sum _{i}E_{i}$ | J | ML$^2$T$^{−2}$ |
| Enthalpy | H | $H=U+pV$ | J | ML$^2$T$^{−2}$ |
| Partition Function | Z | | 1 | 1 |
| Gibbs free energy | G | $G=H-TS$ | J | ML$^2$T$^{−2}$ |
| Chemical potential (of component i in a mixture) | μ$_i$ | $\mu _{i}=\left(\partial U/\partial N_{i}\right)_{N_{j\neq i},S,V}$ $\mu _{i}=\left(\partial F/\partial N_{i}\right)_{T,V}$, where $F$ is not proportional to $N$ because $\mu _{i}$ depends on pressure. $\mu _{i}=\left(\partial G/\partial N_{i}\right)_{T,P}$, where $G$ is proportional to $N$ (as long as the molar ratio composition of the system remains the same) because $\mu _{i}$ depends only on temperature and pressure and composition. $\mu _{i}/\tau =-1/k_{\text{B}}\left(\partial S/\partial N_{i}\right)_{U,V}$ | J | ML$^2$T$^{−2}$ |
| Helmholtz free energy | A, F | $F=U-TS$ | J | ML$^2$T$^{−2}$ |
| Landau potential, Landau free energy, Grand potential | Ω, Φ$_G$ | $\Omega =U-TS-\mu N\$ | J | ML$^2$T$^{−2}$ |
| Massieu potential, Helmholtz free entropy | Φ | $\Phi =S-U/T$ | J⋅K$^{−1}$ | ML$^2$T$^{−2}$Θ$^{−1}$ |
| Planck potential, Gibbs free entropy | Ξ | $\Xi =\Phi -pV/T$ | J⋅K$^{−1}$ | ML$^2$T$^{−2}$Θ$^{−1}$ |

### Thermal properties of matter

| Quantity (common name/s) | (Common) symbol/s | Defining equation | SI unit | Dimension |
| General heat/thermal capacity | C | $C=\partial Q/\partial T$ | J⋅K$^{−1}$ | ML$^2$T$^{−2}$Θ$^{−1}$ |
| Heat capacity (isobaric) | C$_p$ | $C_{p}=\partial H/\partial T$ | J⋅K$^{−1}$ | ML$^2$T$^{−2}$Θ$^{−1}$ |
| Specific heat capacity (isobaric) | C$_{mp}$ | $C_{mp}=\partial ^{2}Q/\partial m\partial T$ | J⋅kg$^{−1}$⋅K$^{−1}$ | L$^2$T$^{−2}$Θ$^{−1}$ |
| Molar specific heat capacity (isobaric) | C$_{np}$ | $C_{np}=\partial ^{2}Q/\partial n\partial T$ | J⋅K$^{−1}$⋅mol$^{−1}$ | ML$^2$T$^{−2}$Θ$^{−1}$N$^{−1}$ |
| Heat capacity (isochoric/volumetric) | C$_V$ | $C_{V}=\partial U/\partial T$ | J⋅K$^{−1}$ | ML$^2$T$^{−2}$Θ$^{−1}$ |
| Specific heat capacity (isochoric) | C$_{mV}$ | $C_{mV}=\partial ^{2}Q/\partial m\partial T$ | J⋅kg$^{−1}$⋅K$^{−1}$ | L$^2$T$^{−2}$Θ$^{−1}$ |
| Molar specific heat capacity (isochoric) | C$_{nV}$ | $C_{nV}=\partial ^{2}Q/\partial n\partial T$ | J⋅K⋅$^{−1}$ mol$^{−1}$ | ML$^2$T$^{−2}$Θ$^{−1}$N$^{−1}$ |
| Specific latent heat | L | $L=\partial Q/\partial m$ | J⋅kg$^{−1}$ | L$^2$T$^{−2}$ |
| Ratio of isobaric to isochoric heat capacity, heat capacity ratio, adiabatic index, Laplace coefficient | γ | $\gamma =C_{p}/C_{V}=c_{p}/c_{V}=C_{mp}/C_{mV}$ | 1 | 1 |

### Thermal transfer

| Quantity (common name/s) | (Common) symbol/s | Defining equation | SI unit | Dimension |
| Temperature gradient | No standard symbol | $\nabla T$ | K⋅m$^{−1}$ | ΘL$^{−1}$ |
| Thermal conduction rate, thermal current, thermal/heat flux, thermal power transfer | P | $P=\mathrm {d} Q/\mathrm {d} t$ | W | ML$^2$T$^{−3}$ |
| Thermal intensity | I | $I=\mathrm {d} P/\mathrm {d} A$ | W⋅m$^{−2}$ | MT$^{−3}$ |
| Thermal/heat flux density (vector analogue of thermal intensity above) | q | $Q=\iint \mathbf {q} \cdot \mathrm {d} \mathbf {S} \mathrm {d} t$ | W⋅m$^{−2}$ | MT$^{−3}$ |

## Equations

The equations in this article are classified by subject.

### Thermodynamic processes

| Physical situation | Equations |
| Isentropic process (adiabatic and reversible) | $Q=0,\quad \Delta U=-W$ For an ideal gas $p_{1}V_{1}^{\gamma }=p_{2}V_{2}^{\gamma }$ $T_{1}V_{1}^{\gamma -1}=T_{2}V_{2}^{\gamma -1}$ $p_{1}^{1-\gamma }T_{1}^{\gamma }=p_{2}^{1-\gamma }T_{2}^{\gamma }$ |
| Isothermal process | $\Delta U=0,\quad W=Q$ For an ideal gas $W=kTN\ln(V_{2}/V_{1})$ $W=nRT\ln(V_{2}/V_{1})$ |
| Isobaric process | p$_1$ = p$_2$, p = constant $W=p\Delta V,\quad Q=\Delta U+p\delta V$ |
| Isochoric process | V$_1$ = V$_2$, V = constant $W=0,\quad Q=\Delta U$ |
| Free expansion | $\Delta U=0$ |
| Work done by an expanding gas | Process $W=\int _{V_{1}}^{V_{2}}p\mathrm {d} V$ Net work done in cyclic processes $W=\oint _{\mathrm {cycle} }p\mathrm {d} V=\oint _{\mathrm {cycle} }\Delta Q$ |

### Kinetic theory

Ideal gas equations :
| Physical situation | Nomenclature | Equations |
| Ideal gas law | p = pressure V = volume of container T = temperature n = amount of substance R = gas constant N = number of molecules k = Boltzmann constant | $pV=nRT=kTN$ ${\frac {p_{1}V_{1}}{p_{2}V_{2}}}={\frac {n_{1}T_{1}}{n_{2}T_{2}}}={\frac {N_{1}T_{1}}{N_{2}T_{2}}}$ |
| Pressure of an ideal gas | m = mass of one molecule M$_m$ = molar mass | $p={\frac {Nm\langle v^{2}\rangle }{3V}}={\frac {nM_{m}\langle v^{2}\rangle }{3V}}={\frac {1}{3}}\rho \langle v^{2}\rangle$ |

#### Ideal gas

| Quantity | General Equation | Isobaric Δp = 0 | Isochoric ΔV = 0 | Isothermal ΔT = 0 | Adiabatic $Q=0$ |
| Work W | $\delta W=-pdV\;$ | $-p\Delta V\;$ | $0\;$ | $-nRT\ln {\frac {V_{2}}{V_{1}}}\;$ $-nRT\ln {\frac {P_{1}}{P_{2}}}\;$ | ${\frac {PV^{\gamma }(V_{f}^{1-\gamma }-V_{i}^{1-\gamma })}{1-\gamma }}=C_{V}\left(T_{2}-T_{1}\right)$ |
| Heat Capacity C | (as for real gas) | $C_{p}={\frac {5}{2}}nR$ (for monatomic ideal gas) $C_{p}={\frac {7}{2}}nR$ (for diatomic ideal gas) | $C_{V}={\frac {3}{2}}nR$ (for monatomic ideal gas) $C_{V}={\frac {5}{2}}nR\;$ (for diatomic ideal gas) | | |
| Internal Energy ΔU | $\Delta U=C_{V}\Delta T\;$ | $Q+W\;$ $Q_{p}-p\Delta V$ | $Q\;$ $C_{V}\left(T_{2}-T_{1}\right)$ | $0\;$ $Q=-W$ | $W\;$ $C_{V}\left(T_{2}-T_{1}\right)$ |
| Enthalpy ΔH | $H=U+pV\;$ | $C_{p}\left(T_{2}-T_{1}\right)\;$ | $Q_{V}+V\Delta p\;$ | $0\;$ | $C_{p}\left(T_{2}-T_{1}\right)\;$ |
| Entropy Δs | $\Delta S=C_{V}\ln {T_{2} \over T_{1}}+nR\ln {V_{2} \over V_{1}}$ $\Delta S=C_{p}\ln {T_{2} \over T_{1}}-nR\ln {p_{2} \over p_{1}}$ | $C_{p}\ln {\frac {T_{2}}{T_{1}}}\;$ | $C_{V}\ln {\frac {T_{2}}{T_{1}}}\;$ | $nR\ln {\frac {V_{2}}{V_{1}}}\;$ ${\frac {Q}{T}}\;$ | $C_{p}\ln {\frac {V_{2}}{V_{1}}}+C_{V}\ln {\frac {p_{2}}{p_{1}}}=0\;$ |
| Constant | $\;$ | ${\frac {V}{T}}\;$ | ${\frac {p}{T}}\;$ | $pV\;$ | $pV^{\gamma }\;$ |

### Entropy

* $S=k_{\mathrm {B} }\ln \Omega$, where k$_B$ is the Boltzmann constant, and Ω denotes the volume of macrostate in the phase space or otherwise called thermodynamic probability.
* $dS={\frac {\delta Q}{T}}$, for reversible processes only

### Statistical physics

Below are useful results from the Maxwell–Boltzmann distribution for an ideal gas, and the implications of the Entropy quantity. The distribution is valid for atoms or molecules constituting ideal gases.
| Physical situation | Nomenclature | Equations |
| Maxwell–Boltzmann distribution | v = velocity of atom/molecule, m = mass of each molecule (all molecules are identical in kinetic theory), γ(p) = Lorentz factor as function of momentum (see below) Ratio of thermal to rest mass-energy of each molecule: $\theta =k_{\text{B}}T/mc^{2}$ K$_2$ is the modified Bessel function of the second kind. | Non-relativistic speeds $P\left(v\right)=4\pi \left({\frac {m}{2\pi k_{\text{B}}T}}\right)^{3/2}v^{2}e^{-mv^{2}/2k_{\text{B}}T}$ Relativistic speeds (Maxwell–Jüttner distribution) $f(p)={\frac {1}{4\pi m^{3}c^{3}\theta K_{2}(1/\theta )}}e^{-\gamma (p)/\theta }$ |
| Entropy Logarithm of the density of states | P$_i$ = probability of system in microstate i Ω = total number of microstates | $S=-k_{\text{B}}\sum _{i}P_{i}\ln P_{i}=k_{\mathrm {B} }\ln \Omega$ where: $P_{i}=1/\Omega$ |
| Entropy change | | $\Delta S=\int _{Q_{1}}^{Q_{2}}{\frac {\mathrm {d} Q}{T}}$ $\Delta S=k_{\text{B}}N\ln {\frac {V_{2}}{V_{1}}}+NC_{V}\ln {\frac {T_{2}}{T_{1}}}$ |
| Entropic force | | $\mathbf {F} _{\mathrm {S} }=-T\nabla S$ |
| Equipartition theorem | d$_f$ = degree of freedom | Average kinetic energy per degree of freedom $\langle E_{\mathrm {k} }\rangle ={\frac {1}{2}}kT$ Internal energy $U=d_{\text{f}}\langle E_{\mathrm {k} }\rangle ={\frac {d_{\text{f}}}{2}}kT$ |
Corollaries of the non-relativistic Maxwell–Boltzmann distribution are below.
| Physical situation | Nomenclature | Equations |
| Mean speed | | $\langle v\rangle ={\sqrt {\frac {8k_{\text{B}}T}{\pi m}}}$ |
| Root mean square speed | | $v_{\mathrm {rms} }={\sqrt {\langle v^{2}\rangle }}={\sqrt {\frac {3k_{\text{B}}T}{m}}}$ |
| Modal speed | | $v_{\mathrm {mode} }={\sqrt {\frac {2k_{\text{B}}T}{m}}}$ |
| Mean free path | σ = effective cross-section n = volume density of number of target particles ℓ = mean free path | $\ell =1/{\sqrt {2}}n\sigma$ |

### Quasi-static and reversible processes

For quasi-static and reversible processes, the first law of thermodynamics is:
    $dU=\delta Q-\delta W$
where δQ is the heat supplied to the system and δW is the work done by the system.

### Thermodynamic potentials

The following energies are called the thermodynamic potentials,
| Name | Symbol | Formula | Natural variables |
| Internal energy | $U$ | $\int \left(T\,\mathrm {d} S-p\,\mathrm {d} V+\sum _{i}\mu _{i}\mathrm {d} N_{i}\right)$ | $S,V,\{N_{i}\}$ |
| Helmholtz free energy | $A$ | $U-TS$ | $T,V,\{N_{i}\}$ |
| Enthalpy | $H$ | $U+pV$ | $S,p,\{N_{i}\}$ |
| Gibbs free energy | $G$ | $U+pV-TS$ | $T,p,\{N_{i}\}$ |
| Landau potential, or grand potential | $\Omega$, $\Phi _{\text{G}}$ | $U-TS-$$\sum _{i}\,$$\mu _{i}N_{i}$ | $T,V,\{\mu _{i}\}$ |
and the corresponding fundamental thermodynamic relations or "master equations" are:
| Potential | Differential |
| Internal energy | $dU\left(S,V,{N_{i}}\right)=TdS-pdV+\sum _{i}\mu _{i}dN_{i}$ |
| Enthalpy | $dH\left(S,p,{N_{i}}\right)=TdS+Vdp+\sum _{i}\mu _{i}dN_{i}$ |
| Helmholtz free energy | $dF\left(T,V,{N_{i}}\right)=-SdT-pdV+\sum _{i}\mu _{i}dN_{i}$ |
| Gibbs free energy | $dG\left(T,p,{N_{i}}\right)=-SdT+Vdp+\sum _{i}\mu _{i}dN_{i}$ |

### Maxwell's relations

The four most common Maxwell's relations are:
| Physical situation | Nomenclature | Equations |
| Thermodynamic potentials as functions of their natural variables | $U(S,V)\,$ = Internal energy $H(S,P)\,$ = Enthalpy $F(T,V)\,$ = Helmholtz free energy $G(T,P)\,$ = Gibbs free energy | $\left({\frac {\partial T}{\partial V}}\right)_{S}=-\left({\frac {\partial P}{\partial S}}\right)_{V}={\frac {\partial ^{2}U}{\partial S\partial V}}$ $\left({\frac {\partial T}{\partial P}}\right)_{S}=+\left({\frac {\partial V}{\partial S}}\right)_{P}={\frac {\partial ^{2}H}{\partial S\partial P}}$ $+\left({\frac {\partial S}{\partial V}}\right)_{T}=\left({\frac {\partial P}{\partial T}}\right)_{V}=-{\frac {\partial ^{2}F}{\partial T\partial V}}$ $-\left({\frac {\partial S}{\partial P}}\right)_{T}=\left({\frac {\partial V}{\partial T}}\right)_{P}={\frac {\partial ^{2}G}{\partial T\partial P}}$ |
More relations include the following.
| $\left({\partial S \over \partial U}\right)_{V,N}={1 \over T}$ | $\left({\partial S \over \partial V}\right)_{N,U}={p \over T}$ | $\left({\partial S \over \partial N}\right)_{V,U}=-{\mu \over T}$ |
| $\left({\partial T \over \partial S}\right)_{V}={T \over C_{V}}$ | $\left({\partial T \over \partial S}\right)_{P}={T \over C_{P}}$ | |
| $-\left({\partial p \over \partial V}\right)_{T}={1 \over {VK_{T}}}$ | | |
Other differential equations are:
| Name | H | U | G |
| Gibbs–Helmholtz equation | $H=-T^{2}\left({\frac {\partial \left(G/T\right)}{\partial T}}\right)_{p}$ | $U=-T^{2}\left({\frac {\partial \left(F/T\right)}{\partial T}}\right)_{V}$ | $G=-V^{2}\left({\frac {\partial \left(F/V\right)}{\partial V}}\right)_{T}$ |
| | $\left({\frac {\partial H}{\partial p}}\right)_{T}=V-T\left({\frac {\partial V}{\partial T}}\right)_{P}$ | $\left({\frac {\partial U}{\partial V}}\right)_{T}=T\left({\frac {\partial P}{\partial T}}\right)_{V}-P$ | |

### Quantum properties

* $U=Nk_{\text{B}}T^{2}\left({\frac {\partial \ln Z}{\partial T}}\right)_{V}$
* $S={\frac {U}{T}}+Nk_{\text{B}}\ln Z-Nk\ln N+Nk$ Indistinguishable Particles
where N is number of particles, h is that Planck constant, I is moment of inertia, and Z is the partition function, in various forms:
| Degree of freedom | Partition function |
| Translation | $Z_{t}={\frac {(2\pi mk_{\text{B}}T)^{\frac {3}{2}}V}{h^{3}}}$ |
| Vibration | $Z_{v}={\frac {1}{1-e^{\frac {-h\omega }{2\pi k_{\text{B}}T}}}}$ |
| Rotation | $Z_{r}={\frac {2Ik_{\text{B}}T}{\sigma ({\frac {h}{2\pi }})^{2}}}$ where: σ = 1 (heteronuclear molecules) σ = 2 (homonuclear) |

## Thermal properties of matter

| Coefficients | Equation |
| Joule-Thomson coefficient | $\mu _{JT}=\left({\frac {\partial T}{\partial p}}\right)_{H}$ |
| Compressibility (constant temperature) | $K_{T}=-{1 \over V}\left({\partial V \over \partial p}\right)_{T,N}$ |
| Coefficient of thermal expansion (constant pressure) | $\alpha _{p}={\frac {1}{V}}\left({\frac {\partial V}{\partial T}}\right)_{p}$ |
| Heat capacity (constant pressure) | $C_{p}=\left({\partial Q_{rev} \over \partial T}\right)_{p}=\left({\partial U \over \partial T}\right)_{p}+p\left({\partial V \over \partial T}\right)_{p}=\left({\partial H \over \partial T}\right)_{p}=T\left({\partial S \over \partial T}\right)_{p}$ |
| Heat capacity (constant volume) | $C_{V}=\left({\partial Q_{rev} \over \partial T}\right)_{V}=\left({\partial U \over \partial T}\right)_{V}=T\left({\partial S \over \partial T}\right)_{V}$ |

### Thermal transfer

| Physical situation | Nomenclature | Equations |
| Net intensity emission/absorption | T$_{external}$ = external temperature (outside of system) T$_{system}$ = internal temperature (inside system) ε = emissivity | $I=\sigma \epsilon \left(T_{\mathrm {external} }^{4}-T_{\mathrm {system} }^{4}\right)$ |
| Internal energy of a substance | C$_V$ = isovolumetric heat capacity of substance ΔT = temperature change of substance | $\Delta U=NC_{V}\Delta T$ |
| Meyer's equation | C$_p$ = isobaric heat capacity C$_V$ = isovolumetric heat capacity n = amount of substance | $C_{p}-C_{V}=nR$ |
| Effective thermal conductivities | λ$_i$ = thermal conductivity of substance i λ$_{net}$ = equivalent thermal conductivity | Series $\lambda _{\mathrm {net} }=\sum _{j}\lambda _{j}$ Parallel ${\frac {1}{\lambda }}_{\mathrm {net} }=\sum _{j}\left({\frac {1}{\lambda }}_{j}\right)$ |

### Thermal efficiencies

| Physical situation | Nomenclature | Equations |
| Thermodynamic engines | η = efficiency W = work done by engine Q$_H$ = heat energy in higher temperature reservoir Q$_L$ = heat energy in lower temperature reservoir T$_H$ = temperature of higher temp. reservoir T$_L$ = temperature of lower temp. reservoir | Thermodynamic engine: $\eta =\left|{\frac {W}{Q_{\text{H}}}}\right|$ Carnot engine efficiency: $\eta _{\text{c}}=1-\left|{\frac {Q_{\text{L}}}{Q_{\text{H}}}}\right|=1-{\frac {T_{\text{L}}}{T_{\text{H}}}}$ |
| Refrigeration | K = coefficient of refrigeration performance | Refrigeration performance $K=\left|{\frac {Q_{\text{L}}}{W}}\right|$ Carnot refrigeration performance $K_{\text{C}}={\frac {|Q_{\text{L}}|}{|Q_{\text{H}}|-|Q_{\text{L}}|}}={\frac {T_{\text{L}}}{T_{\text{H}}-T_{\text{L}}}}$ |